Find the equation the equation of sphere cocentric with sphere `2x^2+2y^2+2z^2-6x+2y-4z=1` and double its radius.

Answer

To find the equation of a sphere that is concentric with the given sphere \(2x^2 + 2y^2 + 2z^2 - 6x + 2y - 4z = 1\) and has double its radius, we can follow these steps:

### Step 1: Rewrite the given sphere equation in standard form
We start with the equation:
\[ 2x^2 + 2y^2 + 2z^2 - 6x + 2y - 4z = 1 \]
To simplify, we can divide the entire equation by 2:
\[ x^2 + y^2 + z^2 - 3x + y - 2z = \frac{1}{2} \]

### Step 2: Rearrange into standard form
We can rearrange this equation to match the standard form of a sphere's equation:
\[ x^2 + y^2 + z^2 + 2ax + 2by + 2cz + d = 0 \]
This gives us:
\[ x^2 + y^2 + z^2 - 3x + y - 2z - \frac{1}{2} = 0 \]
From this, we can identify:
- \(2a = -3 \Rightarrow a = -\frac{3}{2}\)
- \(2b = 1 \Rightarrow b = \frac{1}{2}\)
- \(2c = -2 \Rightarrow c = -1\)
- \(d = -\frac{1}{2}\)

### Step 3: Find the center of the sphere
The center of the sphere in standard form is given by \((-a, -b, -c)\):
\[ \text{Center} = \left(-\left(-\frac{3}{2}\right), -\left(\frac{1}{2}\right), -(-1)\right) = \left(\frac{3}{2}, -\frac{1}{2}, 1\right) \]

### Step 4: Calculate the radius of the sphere
The radius \(r\) can be calculated using the formula:
\[ r = \sqrt{a^2 + b^2 + c^2 - d} \]
Substituting the values we found:
\[ r = \sqrt{\left(-\frac{3}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + (-1)^2 - \left(-\frac{1}{2}\right)} \]
Calculating each term:
\[ = \sqrt{\frac{9}{4} + \frac{1}{4} + 1 + \frac{1}{2}} = \sqrt{\frac{9}{4} + \frac{1}{4} + \frac{4}{4} + \frac{2}{4}} = \sqrt{\frac{16}{4}} = \sqrt{4} = 2 \]
So, the radius of the given sphere is \(2\).

### Step 5: Double the radius
Since we need a sphere that is concentric with the original sphere and has double the radius:
\[ \text{New radius} = 2 \times 2 = 4 \]

### Step 6: Write the equation of the new sphere
Using the center \(\left(\frac{3}{2}, -\frac{1}{2}, 1\right)\) and the new radius \(4\), we can write the equation of the new sphere:
\[ \left(x - \frac{3}{2}\right)^2 + \left(y + \frac{1}{2}\right)^2 + (z - 1)^2 = 4^2 \]
This simplifies to:
\[ \left(x - \frac{3}{2}\right)^2 + \left(y + \frac{1}{2}\right)^2 + (z - 1)^2 = 16 \]

### Final Answer
The equation of the sphere that is concentric with the given sphere and has double its radius is:
\[ \left(x - \frac{3}{2}\right)^2 + \left(y + \frac{1}{2}\right)^2 + (z - 1)^2 = 16 \]